Showing papers on "Recursively enumerable language published in 1975"

Splitting an -recursively enumerable set

Abstract: We extend the priority method in a-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible ca. THEOREM: Let C be a regular car.e. set and D be a nonrecursive cs-r.e. set. Then there are regular or-r.e. sets A and B such that A U B = C, A n B=0, A, B

The disjunction property implies the numerical existence property.

Abstract: Any recursively enumerable extension of intuitionistic arithmetic which obeys the disjunction property obeys the numerical existence property. Any recursively enumerable extension of intuitionistic arithmetic proves its own disjunction property if and only if it proves its own inconsistency.

On the syntactic structures of unrestricted grammars II. Automata

Abstract: We define a generalization of the finite state acceptors for derivation structures and for phrase structures. Corresponding to the Chomsky hierarchy of grammars, there is a hierarchy of acceptors, and for both kinds of structures, the type 2 acceptors are tree automata. For i = 0, 1, 2, 3, the sets of structures recognized by the type i acceptors are just the sets of projections of the structures of the type i grammars, and the languages of the type i acceptors are just the type i languages. Finally, we prove that the set of syntactic structures of a recursively enumerable language is recursive.

A theorem on shortening the length of proof in formal systems of arithmetic

Abstract: This note is concerned with an aspect of the length of proof of formulas in recursively enumerable theories T adequate for recursive arithmetic. In particular, we consider the relative length of proof of formulas in the theories T and T ( S ), where F represents an r.e. set A in T and T ( S ) is the theory obtained from T by adjunction, as a new axiom, of a sentence S undecidable in T . Throughout the sequel T is a consistent, r.e. theory with standard formalization [7] in which all recursive functions of one variable are definable, and in which there is a binary formula x ≤ satisfying the well-known conditions [7]: Here is the constant term corresponding to the natural number n . W n is the n th r.e. set in a standard enumeration of the r.e. sets. Also, we assume an a priori Godel numbering of our formalism satisfying the usual conditions, so that all formulas are numbers ab initio. In the more common applications of the theorem below, if F is a k -ary formula of T , is a natural number that measures in some way the length of the shortest proof of in T .

The decision problem for recursively enumerable degrees

Abstract: If I have any message today for mathematicians in general, it is that consideration of difficult problems can be useful even when the problem is at present beyond solution. The problem I will discuss is unlikely to be solved in the near future, but I hope to show how the study of it leads to many more accessible problems. In order to state the problem, we need some definitions. To save words, we agree that number means natural number (nonnegative integer) and set means set of numbers. A set A is recursive if there is an algorithm for determining whether any given number is in A. A set A is recursive in a set B if there is an algorithm by which we can decide whether any given number x is in A, provided we are supplied with answers to all questions we choose to ask of the form Ts y in BT. As an example, let A ={2x : x e B}. Then B is recursive in A ; for x e B iff 2x € A. Also A is recursive in B ; for x e A iff x is even and \\x e B. (All this is independent of the choice of B.) Writing A^RB for A is recursive in B, we easily see that (1) A ^ R A ,

Combinatorial Systems II, Non-Cylindrical Problems

Abstract: Publisher Summary Certain problems associated with various kinds of combinatorial systems are always cylinders and therefore of maximum one-one degree within their many-one degrees. This chapter identifies some combinatorial problems that can be noncylinders. The purpose of the chapter is to prove the theorem which explains that every nonrecursive, recursively enumerable one–one degree can be represented by a special confluence problem of a system function.

A note on the recursive enumerability of some classes of recursively enumerable languages : (prepublication)

Abstract: Abstract An elementary proof is presented for the fact that the class of infinite recursive languages is not recursively enumerable. Its relevance for contemporary linguistics and computer science is explained.

Embedding relations in the lattice of recursively enumerable sets

Abstract: Let ℰ* be the lattice of recursively enumerable sets of natural numbers modulo finite differences We characterize the relations which can be embedded in ℰ* by using certain collections of maximal sets as domain and using Lachlan's notion of major subsets to code in the relation in certain natural ways We show that attempts to prove the undecidability of ℰ* by using such embeddings fail